Mathematics

Solve $$\int {\sqrt {\dfrac{{1 + x}}{{1 - x}}} } $$


SOLUTION

$$\int {\sqrt {{{1 + x} \over {1 - x}}} dx} $$

$$put\,x = \cos 2\theta $$

$$d\theta  = 12{\sin ^2}\theta \,d\theta $$

$$\int {\sqrt {{{1 + {{\cos }^2}\theta } \over {1 - {{\cos }^2}\theta }}}  \times 2{{\sin }^2}\theta \,d\theta } $$

$$\int {\sqrt {{{2{{\cos }^2}\theta } \over {2{{\sin }^2}\theta }}}  \times  - 2{{\sin }^2}\theta \,d\theta } $$

$$ - 2\int {{{\cos \theta } \over {\sin \theta }} \times 2\sin \theta \cos \theta \,d\theta } $$

$$ - 4\int {{{1 + {{\cos }^2}\theta } \over 2}d\theta } $$

$$ - 2\left( {\theta  + {{{{\sin }^2}\theta } \over 2}} \right) + C$$

$$ - 2\theta  - {\sin ^2}\theta  + C$$

$$ - {\cos ^1}x - \sqrt {1 - {x^2}}  + c$$

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Subjective Medium Published on 17th 09, 2020
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